WASHINGTON — President Trump’s decision to impose tariffs on imported steel and aluminum has prompted a stampede by foreign countries and companies and their American partners pressing for exemptions and exclusions that could be worth billions of dollars in trade.

Corporations and foreign leaders are leaning on personal relationships in vying for meetings with White House officials, hiring lawyers and lobbyists to defend them, and drafting messages to persuade the public of the importance of free trade.

On Thursday afternoon, top oil and gas executives left an American Petroleum Institute board meeting at the Trump International Hotel in Washington and headed to the White House to personally lobby Mr. Trump, telling him that his pending steel tariffs could raise the cost of pipelines and extraction projects across the country, and discussing possible exemptions.

Three days earlier, the American Soybean Association, whose members are among the biggest United States exporters, requested a meeting with Mr. Trump. After hearing nothing, they sent 120 farmers and association employees to meet with lawmakers from farming states, arguing that the tariffs could set off retaliation and hurt American agriculture.

The rush started when Mr. Trump floated the potential for exclusions, saying that Canada and Mexico would be exempted from the 25 percent tariffs on steel and 10 percent tariffs on aluminum as the countries renegotiate the North American Free Trade Agreement. He also spoke warmly of Australia as an ally and trading partner, and said that other nations with close relationships to the United States might be exempted, too.

“The dinner bell is ringing for the trade bar and associated lobbyists and consultants,” said Chip Roh, a former partner at Weil, Gotshal & Manges. Lawyers and lobbyists are employed on both sides, arguing for and against exemptions, he said, adding, “It creates a fertile field.”

But the Trump administration has yet to clarify what precisely will qualify trading partners for an exemption, other than a vague reference to protecting America’s national security and reducing its trade deficit. That has led to a chaotic and anxious period as companies and countries try their best to meet unclear specifications, and the clock continues to tick toward tariffs that are expected to go into effect next Friday, wiping out valuable contracts and putting global supply chains at risk.

Robert Lighthizer, the government’s top trade negotiator, has been charged with helping decide which countries will be excluded, while Wilbur Ross, the commerce secretary, has been tasked with a parallel process for exempting imports of certain products that are not widely available in the United States. The Commerce Department has developed its procedures for such exclusions, but it has not yet released them publicly. Those will be issued by Sunday, a spokesman said.

The European Union commissioner for trade, Cecilia Malmstrom, speaking on Wednesday to the European Parliament, said she left a meeting with Mr. Lighthizer over the weekend in Brussels without “sufficient clarity” about how countries would secure exclusions.

In the interim, trade lawyers and lobbyists in Washington have begun offering assistance in securing exemptions and exclusions for foreign countries, companies and their United States partners. Lobbyists said they intended to seek support from elected officials in pushing for exemptions, and some lawmakers have already weighed in.

Senator Lisa Murkowski, Republican of Alaska, has already helped make the case for an exemption for a major pipeline project in her state, predicting that the tariffs could add $500 million to the cost of a $43 billion natural gas export terminal that her state is building with China.

The American Petroleum Institute sent 12 top executives from energy companies including BP America, Exxon Mobil, Phillips 66 and Shell to lobby Mr. Trump and Vice President Mike Pence. The assembled group explained that free trade has been a boon to the industry and stressed the importance of steel to its operations, according to a person briefed on the meeting.

Mr. Trump in turn told the executives that his administration wanted to make sure that the tariffs did not hurt the industry, the person briefed on the meeting said.

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Soybeans being sorted at the Peterson Farms Seed plant in Fargo, N.D. Soybeans are one of the United States’ biggest exports.CreditDan Koeck/Reuters

The trade group issued a statement on Thursday saying that its executives “highlighted a host of the industry’s priority issues, including the importance of trade policies that recognize the integrated nature of North American and global markets.”

In recent days, Hogan Lovells and Covington & Burling — firms that specialize in international trade law and lobbying — published memos heralding their ability to help clients obtain exemptions, and dropping hints as to how they might go about doing so.

Hogan Lovells noted that existing contracts for steel or aluminum imports could potentially receive “grandfather” exemptions. One company whose costs could rise because of the tariffs, Kinder Morgan, a pipeline maker based in Houston, recently signed Hogan Lovells to lobby on the impact of the “imposition of steel tariffs on pipeline construction,” according to a congressional filing posted last week.

CGCN Group, a lobbying firm, recommends that companies highlight the advantages of global trade to a public that has grown skeptical, according to polls.

“Proponents of free trade need to tailor their arguments to those Americans who have watched factories close and wages stagnate,” according to a memo that CGCN sent last week to clients and prospective clients. It suggested that companies seeking exemptions refrain from warning of tumbling stock prices or potential layoffs, and instead argue that “the more money companies spend on steel and aluminum, the less they have to hire workers or give them a raise.”

Daniel Porter, chief trade lawyer for the Japan Iron and Steel Federation, said he was counseling his clients that the exemption process could look similar to previous tariff processes, which allowed companies to file for exemptions and then gave their American competitors a chance to object to any of those requests.

With the current steel and aluminum tariffs, however, there is likely to be one notable change: Only companies based in the United States will be allowed to file for exclusions. That means steel makers in Japan and Europe will need to get their American customers to come forward and file the request on their behalf.

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A Union Pacific freight train last year in Roxana, Ill. The railroad uses Japanese steel on some of its most heavily traveled tracks.CreditLuke Sharrett/Bloomberg

Companies like California Steel Industries, which makes flat rolled steel, and the Union Pacific railroad said they depended on foreign steel and were bracing for higher prices if they could not secure exemptions.

Lance M. Fritz, the chief executive of Union Pacific, said the railroad’s most heavily trafficked routes were made with steel from Nippon Steel of Japan, which is harder and more durable than products available in the United States. If Japan does not secure an exemption, Mr. Fritz said, Union Pacific planned to apply for a product exclusion.

Foreign metal makers are also worried that American mills will push aggressively against these exclusions, cutting off products that American companies buy.

“There is that classic gray middle, where the customers say the U.S. mill can’t do an adequate job, and the U.S. mill says, ‘No, you’re being too finicky,’” Mr. Porter said. “Even with the back door, some very real American manufacturing entities are going to be hurt by this.”

That fight is already on display, as American metal producers press the administration to avoid giving in to metal users, like food packagers and beer companies, who are agitating for exemptions.

On Tuesday, the presidents of four trade associations — the American Beverage Association, the Beer Institute, the Brewers Association and the Can Manufacturers Institute — sent a letter to Mr. Ross asking for more clarification on the timing and process for exemptions. They argued that a delay would undoubtedly increase costs for manufacturers and could lead to unintended consequences.

A separate letter on Thursday from the Alliance for American Manufacturing, which supported the tariffs, urged Mr. Ross and Mr. Lighthizer not to undermine them by issuing country exemptions and product exclusions that were too broad. The group said that new steel jobs in Granite City, Ill., and Lorain, Ohio, had already been announced in response to the tariffs and that overly broad exemptions could undermine that progress.

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President Trump welcoming Crown Prince Mohammed bin Zayed Al-Nahyan, the effective ruler of the United Arab Emirates. The Emirates is a top exporter of aluminum to the United States.CreditAl Drago for The New York Times

Foreign governments are also looking for ways to avoid being hit by the tariffs, including having top officials speak personally to Mr. Trump.

“Everybody is going to be calling in their chits and trying to make their points about why their exemptions make sense, but not everyone will be successful,” said Danny E. Sebright, the president of the U.S.-U.A.E. Business Council, a nonprofit group that seeks to bolster trade between the United States and the United Arab Emirates.

The United Arab Emirates is a top exporter of aluminum to the United States. Mr. Sebright said that he expected Crown Prince Mohammed bin Zayed Al-Nahyan, the effective ruler of the Emirates, to visit the United States in the coming weeks, and that aluminum would be a topic of conversation.

“They’re going to work with the Trump administration on this, there is not going to be retaliation,” Mr. Sebright said.

Australia has trumpeted an apparent exemption from Mr. Trump’s tariffs, which an Australian government official said on Monday was achieved as a result of “high level and sustained advocacy” by some of its highest-ranking officials. The country’s interlocutors included Greg Norman, a pro golfer and friend of the president who signed a letter telling him that the tariffs could have a negative effect on the relationship between the United States and Australia.

Other governments plan to do less lobbying and more threatening.

David O’Sullivan, the European Union ambassador to the United States, said that the bloc was “happy to explain why European exports are not the problem.” He added, “But we are not open to or willing to entertain links to unrelated issues.”

On Friday, the European Union published a 10-page list of American products that would be targets for retaliation if the steel and aluminum tariffs go through, including peanut butter, grains, motorcycles and steel.